Algorithms Of Fixed Points For Solutions Of Some Kinds Of Generalized Variational Inequalities

In this PhD thesis,it is investigated that the algorithms of iterative approximate fixed points for solutions of some kinds of generalized variational inequalities in Hilbert spaces and Banach spaces,respectively.In the real iterative approximate process,the existence and approximation problems of solutions for some kinds of variational inequalities are studied by using Mann-Ishikawa method,Halpern method,hybrid method,successive approximation method,and their new modified methods,which are relating to the modern analysis theories of Banach Geometry,Critical Point Theory,Variational Methods,Nonlinear Approximate Theory in Banach Spaces,and Fixed Point Theory and combining the tools of metric projective operator,generalized projective operator,resolvent operator equation etc.The main contents of this thesis are in the following:1.In Chapter 3,firstly,we use the metric projective operator to study iterative approximations for finding a common element of the fixed points of a non-expansive mapping and the set of solutions of a variational inequality for an inverse-strongly monotone mappings in Hilbert space.The conditions which guarantee strong convergence and stability of these approximations with respect to perturbations of non-expansive operator S,metric projection operator P_Ωand constraint setΩare considered.We show that the iterative sequence{x_n} converges strongly to a common element of the two sets of the fixed points and the solutions.Secondly,we obtain a new strong convergence theorem of a new iterative sequence{x_n} on the solutions of a variational inequality for non-expansive mappings in the uniformly convex and smooth Banach spaces also by using the metric projection.2.In Chapter 4,one of main purposes is to establish strong convergence theorem for asymptotically pseudo-contractions in Banach spaces by using the demi-closedness principle, generalized projective operator in the Banach space and the hybrid method in mathematical programming.Another purpose of this chapter is to establish strong convergence theorem for a common fixed point of a finite family of relatively non-expansive mappings in a Banach space by using the generalized projective operator and the hybrid method in mathematical programming. The third purpose of this chapter is that we study the strong convergence of fixed points iteration process for the non-self mapping T:G(?)B→B,which is called the asymptotically weak contractive mapping,by using the brand-new generalized projection method and modified successive approximation methods or the modified Mann iterative sequence method in a Banach spaceB.3.In Chapter 5,firstly,we establish strong convergence theorems for a infinite family of Lipschitz pseudo-contractions in Hilbert spaces by proposing three kinds of new algorithms.Secondly,we study the existence and approximation problem of solutions for the infinite family of generalized set valued quasi-variational inclusions in Banach spaces by using resolvent operator equation.